Numerous authors [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} have obtained lower bounds for \(\gamma (\alpha ,\beta )\)  in different regimes. The most relevant for our discussion is the following result of Héra, Shmerkin, and Yavicoli, which used a variant of Theorem REF  due to Bourgain [8]} to obtain bounds on \(\gamma (\alpha ,2\alpha )\) .
